Lyapunov exponents and all that A. Pikovsky Deapartment of Physics and Astronomy, Potsdam University URL: www.stat.physik.uni-potsdam.de/ pikovsky
Alexandr Mikhailovich Lyapunov * 1857 Jaroslavl 1870 1876 Nizhni Novgorod 1876 1882 St Petersburg University 1884 Magister Thesis (superwiser P. L. Tschebyschev) 1885 1902 Kharkov University 1892 Doctor Thesis ( A general problem of stability of motion ) 1902 1917 St. Petersburg University 1918 Odessa 1
Elementary one-dimensional dynamics One-dimensional map and its linearization lim T x n+1 = F(x n ) ln δx T ln δx 0 T δx n+1 = F (x n )δx n ln δx n+1 = ln F (x n ) +ln δx n ln δx T ln δx 0 = = lim T 1 T T 1 k=0 T 1 k=0 ln F (x n ) ln F (x n ) = ln F (x n ) = λ 2
Matrix products and Oseledets theorem In dimensions larger than one we have a product of matrices δx n+1 = M n δx n Oseledets multiplicative ergodic theorem: Lyapunov exponents = exponential asymptotic growth rates of vectors χ + (v) = lim T 1 T ln T 1 k=0 M k v are well defined a.e. and attain at most dimm different values. Furthermore, there is a Lyapunov decomposition into subspaces corresponding to the vatious Lyapunov exponents, whose dimension defines the multiplicity of the corresponding exponent. 3
Comparing to other characteristics of stability Lyapunov exponents heavily rely on the existense of ergodic measure, they describe stability in a statistical sense. Remarkable: LEs are not realy needed for mathematical theory of hyperbolic systems (in the book of Katok and Hasselblatt they appear only in an Appendix devoted to non-uniform hyperbolicity) 4
LEs as a main tool of exploring chaos sensitive dependence on initial conditions = positive Lyapunov exponent KS-entropy = sum of positive LEs (Pesin theorem) independent of the metric used (but this is not true in infinite-dimensional case!) invariant to smooth transformations of variables (diffeomorphisms) (but not to general transformations homeomorphisms!) 5
Geometric picture Lyapunov exponents are associated with Lyapunov vectors and with stable and unstable manifolds 6
Numerical implementation Bennetin, Galgani, Giorgilli and Strelcyn (1980) algorithm: Gram- Schmidt orthogonalization Eckmann and Ruelle (1985): QR-decomposition (may be numerically preferable) Greene and Kim (1987): Calculation of LEs and Lyapunov directions Bridges and Reich (2001): A stable variant of continuous-time decomposition 7
Finite-time fluctuations of LEs Calculation of LEs over a time interval T gives a distribution P T (Λ) for which one assumes an ansatz P T (Λ) e TS(Λ) with an entropy function S(Λ) 0 that reaches the maximum at Λ = λ. Another representation via generalized exponents L(q) = lim T ln v q T λ = dl(q) dq q=0 8
Flinite-time fluctuations of negative LE and SNA If the largest LE is negative but the entropy function S(Λ) has a tail at positive Λ: Stability in average but unstable trajectories are inserted in the attractor 4.0 2.0 x 0.0 2.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 θ 9
Coupling sensitivity of chaos Lyapunov exponents in weakly coupled systems depend on the coupling in a singular way ε λ 1 logε 0.04 0.30 0.20 0.02 λ i Λ i (i = 1,2) 0.00 0.02 (λ i Λ i )/σ 2 (i = 1,2) 0.10 0.00 0.10 0.20 0.04 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 ε 0.30 0.05 0.10 0.15 0.20 0.25 1/ ln(ε/σ 2 ) 10
Repulsion of LEs in random systems Lyapunov exponents in noncoupled chaotic systems avoided crossing at ε=10 5 λ i (i=1,...,8) λ i (i=1,...,8) τ 11
Distribution of the Lyapunov exponent spacing has a strong depletion for small λ 1.0 10 0 cumulative distribution function 0.5 ε 0 =0 ε 0 =10 10 ε 0 =10 5 ε 0 =10 3 cumulative distribution function 10 1 10 2 ε 0 =10 10 ε 0 =10 5 ε 0 =10 3 0.0 0.00 0.05 0.10 spacing between Lyapunov exponents λ 100 50 0 1/ λ 12
Conditional/transversal LEs and synchronization conditions Stability of symmetric sets with respect to particular perturbations that break the symmetry is described by linear systems of the type v = M(t)v with a chaotically time-varying matrix M(t) 13
Example: an ensemble of identical systems x k, k = 1,...,N coupled via mean fields g(x) and global variables y: ẋ k = F(x k,y,g;ε), ẏ = G(y,g;ε), Full synchrony x k = x is described by a low-order system ẋ = F(x,y,g;ε), ẏ = G(y,g;ε), Split or evaporation LEs describing the stability of the synchronous cluster are determined in thermodynamic limit N dδx dt = F x δx 14
LEs in noisy systems and sycnhronization by common noise Lyapunov exponents in noisy systems characterize sensitivity to initial conditions for the same realization of noise Synchronization by common noise: largest LE negative: synchronization to a common identical state largest LE positive: desynchronization Example: reliability of neurons under repititions of the same noise 15
Nonlinear exponents Fix the level of the perturbation ε = v and calculate the time to reach the level 2ε: This gives the level-dependent exponent λ(ε) Typically λ(ε) decreases for large ε and λ(0) = λ 16
Lyapunov Exponents in extended systems Take a large finite system of length L, calculate the LEs and look how they change with increase of L: A spectrum f of LEs λ k = f(k/l) which defines the density of the KS-entropy and of the Lyapunov dimension Take an infinite system: Different metrics are not equivalent Physically: a perturbation may not simply grow but come from remote parts of the system 17
Velocity-dependent exponent: Take a local perturbation v(x 0,0) and follow it along the constant-velocity rays: v(x 0 +Vt,t) e tλ(v) Chronotopic Lyapunov analysis: Take a perturbation that is exponential in space v e µx and calculate its LE λ(µ). Velocity-dependent exponent is the Legendre transform of the chronotopic one Lyapunov-Bloch exponent: Take a system of size L and calculate the Lyapunov exponent λ(κ) of the perturbation v e iκx. It determines stability of space-periodic states Statistics of Lyapunov vectors in extended systems may be nontrivial, in many cases it seemingly belongs to a Kardar-Parisi-Zhang universality class